Solving Rational Equations

Front

What is a rational equation?

Back

A rational equation is an equation that involves at least one rational expression, which is a fraction where the numerator and denominator are polynomials.

Front

How do you solve a rational equation?

Back

To solve a rational equation, find a common denominator, eliminate the fractions by multiplying both sides by that denominator, and then solve the resulting polynomial equation.

Front

What is the first step in solving the equation \( \frac{1}{x-2} = \frac{3}{x+1} \)?

Back

The first step is to cross-multiply to eliminate the fractions: \( 1 \cdot (x+1) = 3 \cdot (x-2) \).

Front

What does it mean to check for extraneous solutions?

Back

Checking for extraneous solutions means substituting the found solutions back into the original equation to ensure they do not make any denominator zero.

Front

Solve for x: \( \frac{2}{x} + 3 = 5 \)

Back

x = 1.

Front

What is a common mistake when solving rational equations?

Back

A common mistake is forgetting to check for restrictions on the variable that could make the denominator zero.

Front

Solve for x: \( \frac{x-1}{x+2} = 2 \)

Back

x = 2.

Front

What is the restriction in the equation \( \frac{1}{x-3} = 4 \)?

Back

x cannot be 3, as it would make the denominator zero.

Front

How do you find the least common denominator (LCD) for the fractions \( \frac{1}{x} \) and \( \frac{1}{x-1} \)?

Back

The LCD is \( x(x-1) \).

Front

Solve for x: \( \frac{3}{x+1} - \frac{2}{x-1} = 0 \)

Back

x = 1.